一、主講人介紹：Dario A. Bini

Dario A. Bini，意大利比薩大學數學院教授，主要從事馬爾可夫鏈及排隊問題數值解、矩陣方程數值解法、結構化矩陣計算、幾何矩陣均值及其算法。Dario A. Bini教授在Numerische Mathematik，Mathematics of Computation，SIAM Journal on Scientific Computing，SIAM Journal on Matrix Analysis and Applications，IMA Journal of Numerical Analysis，Numerical Linear Algebra with Applications等計算數學國際頂尖和權威期刊發表論文200余篇，出版Numerical Solution of Algebraic Riccati Equations，Numerical Methods for Structured Markov Chains等計算數學論著7余篇。曾擔任SIAM J. Matrix Analysis Appl.，Electronic Transactions on Numerical Analysis，Electronic Journal of Linear Algebra等計算數學國際頂尖以及權威期刊編委。

二、講座信息

講座摘要：

We consider the problem of solving matrix equations of the kind A_1 X^2+A_0X+A_(-1)=X , where the coefficients  A_r ,r=-1,0,1, are matrices having specific structures, and X is the unknown matrix. The solution of interest is the one that has some minimality properties, say, it has a minimal spectral radius or has nonnegative entries with minimal value. This kind of problem is encountered in the solution of Quasi-Birth-Death processes, a general framework that models real-world problems in terms of Markov chains. In this talk, after presenting and motivating the interest of this class of equations, we investigate some computational issues encountered in their solution. For this class of problems, the coefficients A_r ,r=-1,0,1 ,  are semi-infinite Quasi-Toeplitz (QT) matrices. We give conditions under which the class of QT matrices is a Banach algebra, that is, a vector space closed under multiplication, endowed with a norm that makes it a Banach space. We give conditions under which the sought solution, say the minimal nonnegative one, is still a QT matrix, and describe and analyze  algorithms for its effective computation. Finally, by means of some numerical experiments performed with the CQT Matlab Toolbox, we show the effectiveness of our algorithms。

講座時間：5月26日（星期四）13:30-14:30

騰訊會議號：142-518-059

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國際合作與交流處

數學科學學院  

2022年5月18日